Pattern-avoiding permutations: The case of length three, four, and five

A shorter permutation of length k is said to appear as a pattern in a longer per-mutation of length n if the longer permutation has a subsequence of length k that is order isomorphic to the shorter one. Otherwise, the longer permutation avoids the shorter one as a pattern. We use Sn(τ) to denote the set of permutations of length n that avoid pattern τ. Pattern avoidance induces an equivalence relation on the pattern set Sk. For ρ, τ ∈ Sk, we define the equivalence relation as follows: ρ ∼W τ if and only if |Sn(ρ)| = |Sn(τ)| for all n ≥ 1. The equivalence classes of this relation are called Wilf classes. The main questions are determining the Wilf classes of Sk and enumerating each class. We first study the Wilf classification and enumeration of each class for S3 and S4. We then present some new numerical results regarding the Wilf classification of pairs of patterns of length five. We define a Wilf class as small if it contains only one pair and big if it contains more than one pair. We show that there are at least 968 small Wilf classes and at most 13 big Wilf classes.